Stochastic evaluation methods of a multi-state system via a modular decomposition

Theoretical examinations about multi-state systems started around the publication of Barlow and Proschan [1], which has totally summarised the achievements about binary state systems by many researchers. Recently, a model of a multi-state system is tried to be applied for solving real problems and some authors have published books written from a practical point of view. See Natvig [16], Lisnianski and Levitin [9] and Lisnianski et al. [14], in which we may find real application of the results. In the reliability theory, evaluation of reliability performances of real systems and developing effective evaluation methods are important theme of reliability researchers. In this paper, we propose some evaluation methods of multi-state systems, using basic elements corresponding to the minimal path and cut state vectors of the binary state systems. We especially show convenient stochastic bounds via a modular decomposition. State spaces are assumed to be partially ordered sets, which means that the model proposed in this paper is one of the extremely extended models from the binary case.